On unbounded and binary parameters in multi-parametric programming: applications to mixed-integer bilevel optimization and duality theory

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine cons...

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Bibliographic Details
Published in:Journal of global optimization Vol. 69; no. 3; pp. 587 - 606
Main Authors: Oberdieck, Richard, Diangelakis, Nikolaos A., Avraamidou, Styliani, Pistikopoulos, Efstratios N.
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2017
Springer
Springer Nature B.V
Subjects:
Combinatorial analysis
Computer Science
Formulations
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Optimization
Quadratic programming
Real Functions
Mixed-integer bilevel programming
Multi-parametric programming
Unbounded problems
Binary parameters
ISSN:0925-5001, 1573-2916
Online Access:Get full text
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Summary:In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.
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ISSN:0925-5001
1573-2916
DOI:10.1007/s10898-016-0463-z